Slewing Maneuver of a Flexible Spacecraft Using Finite Time Control Mahmut Reyhanoglu Physical Sciences Department Embry-Riddle Aeronautical University Daytona Beach, FL 32114, USA
[email protected]
Abstract— The slew-maneuver control problem is studied for a flexible spacecraft consisting of a rigid main body to which a long flexible appendage is attached. A nonlinear dynamical system model is first developed using a distributed parameter modeling technique. A filtered finite time control law is then introduced to achieve fast and precise slewing maneuvers without residual vibrations. The results are applied to a benchmark flexible system and a simulation example is included to illustrate the effectiveness of the proposed control technique.
I. INTRODUCTION An important area where flexible structure control finds a widespread use is in control of aerospace systems. An extensive amount of research has been done in the area of active vibration control in aerospace structures such as large flexible satellites, space antennae, space telescopes, and the International Space Station. Other examples of flexible structures include light-weight robot arms and pointing systems. The book by Junkins and Kim [11] addresses in detail the problems associated with modeling and control of flexible structures. Infinite-dimensional models of flexible structures are treated using both Lagrangian approach and extended Hamilton’s principle. The two most common approximate methods for finite dimensional models, the assumed modes method and the finite element method, are thoroughly discussed. The book by Bryson [1] treats extensively several linear control design methods, including LQR (Linear Quadratic Regulator) and LQG (Linear Quadratic Gaussian) methods, for flexible spacecraft. The papers [13]-[15] provide mathematical models and control methodologies for multibody flexible space structures. In the literature, there have been many control schemes proposed for rotational maneuvers of flexible spacecraft with simultaneous vibration suppression. Control methods based on input command pre-shaping and/or time-delay filtering can be found in [5], [25]-[27], and references therein. Boundary feedback control laws are proposed in [16], [20]-[22]. These control laws employ Lyapunov theory for distributed parameter systems, i.e. systems described by partial differential equations; and the stability achieved by these laws holds for the original infinite dimensional systems. Several other types of controllers have been developed for flexible spacecraft, including variable structure controllers ([7], [8], [24]), controllers based on genetic algorithms [6], fuzzy logic controllers [23], structurally stable controllers based k,(((
on internal model principle [19], bang-off-bang control [17], output feedback control [2]-[4], and dynamic dissipative control [9]-[10]. Comparison of various controller designs for an experimental flexible structure can be found in [18]. The organization of the paper is as follows: Section 2 presents the mathematical development of a model for a generic flexible system. The filtered finite time controller design is presented in Section 3. Section 4 introduces a benchmark flexible system model. Simulation results for the model are presented in Section 5. Finally, Section 6 consists of a summary of the paper and concluding remarks about future research. II. M ATHEMATICAL M ODEL A typical spacecraft structure consists of two principal parts: a main body and flexible appendages. The main body of the spacecraft contains all the payload instrumentation and control hardware. Its structure must be rigid in order to withstand mechanical loads during the launch stage. The second part of the spacecraft structure consists of large flexible appendages, such as solar arrays and antennae, built from light materials in order to minimize their weight. These flexible appendages induce structural vibrations that interfere strongly with the rigid-body attitude dynamics. In order to achieve high precision attitude demands, the dynamic effects of flexible appendages have to be taken into account. This section is primarily concerned with the mathematical modeling of a spacecraft consisting of a rigid central body to which a flexible beam-like appendage is attached. The appendage is clamped to the rigid body at one end and free at the other end. We assume that the flexible beam performs only planar motion. The appendage is considered to be a uniform flexible beam, and we make the Euler-Bernoulli assumptions of negligible shear deformation and negligible distributed rotary inertia. Any effects from atmospheric drag are also ignored. The most widely used modeling techniques include distributed parameter modeling, discrete parameter modeling, and finite element modeling techniques. This paper briefly describes the distributed parameter modeling technique. For full details on the modeling of flexible structures, the reader is referred to [11]. Consider a spacecraft model consisting of a rigid body and a single flexible panel modeled as an Euler-Bernoulli beam as shown in Fig. 1. Let XY Z denote the inertial axes
and xyz be the body-fixed axes. The angle θ represents the attitude of the spacecraft and τ is the control torque about the z axis. Let ρl denote the mass per unit length of the beam, Iz denote the rotational inertia of the rigid body about the z axis, and EI be the uniform flexural rigidity of the beam. Then the kinetic energy T and potential energy U can be expressed as: 1 1 T = Iz θ˙2 + 2 2 U=
1 EI 2
0
l
l
Y y
τ y(x,t)
x
r
ρlr˙ 2 dx,
Θ
0
X
(y )2 dx,
where y or y(x, t) denotes the deformation from the rigid body axis at a point x units along the beam and at time t, l is the length of the beam, θ denotes the attitude angle, and r˙ 2 is the square of the velocity of the beam at point x, which is given by ˙ 2 + [(x + l0 )θ˙ + y] ˙ 2, r˙ 2 = (y θ) where l0 is the distance from the spacecraft center of mass to the point of attachment of the beam. We use the assumed mode method to separate the variables for y(x, t): N φk (x)qk (t), y (x, t) =
Fig. 1.
Model of a rigid body with a flexible attachment.
where c is the damping constant for the beam. Consequently, Lagrange’s equations of motion can be written as It + mq q 2 θ¨ + mθq q¨ + 2mq q q˙θ˙ = τ, mq q¨ + mθq θ¨ − mq q θ˙2 + kq + cq˙ = 0.
(7) (8)
k=1
where φk (x) is the kth shape function and qk (t) is the generalized coordinate corresponding to the kth vibrational mode. As in [11], we use the following shape functions 2 kπx kπx 1 k+1 φk (x) = 1 − cos . + (−1) l 2 l It can be shown that these functions satisfy both the geometric (y(0, t) = 0, y (0, t) = 0) and physical boundary conditions (y (l, t) = 0, y (l, t) = 0), and thus they are comparison functions. To obtain the simplest model, only the first mode will be considered, so N = 1. The Lagrangian function L = T − U then can be expressed as 1 1 1 1 L = It θ˙2 + mq q 2 θ˙2 + mq q˙2 + mθq θ˙q˙ − kq 2 , (1) 2 2 2 2 where l mq = ρl φ2 dx, (2) 0 l (x + l0 )φ dx, (3) mθq = ρl 0 l k = EI (φ )2 dx, (4) 0
It = Iz + ρl [(l + l0 )3 − l03 ]/3.
(5)
The structural damping is included via the Rayleigh dissipation function given by 1 (6) R = cq˙2 , 2 k,(((
Using a partial feedback linearization, we obtain θ¨ = u, q¨ + 2ζωn q˙ +
(9) ωn2 q
˙2
= −αu + q θ ,
(10)
where τ − mθq q θ˙2 + αkq + αcq˙ − 2mq q q˙θ˙ , It − αmθq + mq q 2 mθq α= , mq k ωn = , mq c . ζ= 2 mq k u=
(11) (12) (13) (14)
Here u is the new control input, α is the coupling constant, ωn is the natural frequency, and ζ is the damping ratio for the flexible appendage. III. C ONTROLLER D ESIGN In general, to control the rigid body motion, a proportional-plus-derivative (PD) controller is utilized. In this paper, we propose a finite time controller for the system. It is demonstrated that such control can be combined with a notch filter to insure that the inputs at the vibration frequencies are not passed through the filter.
A. Notch Filter A notch filter, also known as a band-reject filter, is a specialized form of a second-order filter [28]. A generalized format of these filters is given by
s2 ωz2 + 2ζz s/ωz + 1 u = 2 2 , (15) v s ωp + 2ζp s/ωp + 1 where u is the filter output, v is the filter input; and ωz , ωp , ζz , and ζp are filter parameters. Different choices and combinations for these parameters yield different filter designs such as lead/lag and all-pass filters as well as the notch filter. For the notch filter in this paper, the parameters are chosen as ωz = ωp = ωn , ζz = ζ, and ζp = 1. Making these substitutions into the original generalized format yields the notch filter u = v
s + 2ζωn s + ωn2 . s2 + 2ωn s + ωn2
Clearly, the control u for the system is the filter output given by ˙ u = z¨ + 2ζωn z˙ + ωn2 z = v − 2ωn (1 − ζ) z. Define the state variables (x1 , x2 , x3 , x4 , x5 , x6 ) = ˙ q, q, (θ, θ, ˙ z, z) ˙ so that the closed-loop system (including the nonlinearity θ˙2 q) can be written as: x˙ 1 = x2 , x˙ 2 = v − 2ωn (1 − ζ)x6 ,
(18) (19)
x˙ 3 = x4 ,
(20)
x˙ 4 =
(x22 −ωn2 )x3 −2ζωn x4 −α[v−2ωn(1−ζ)x6 ],
(21)
x˙ 5 = x6 ,
(22)
x˙ 6 = −2ωn x6 − ωn2 x5 + v,
(23)
where v = −|x1 |a sign(x1 ) − |x2 |b sign(x2 ).
2
(16)
(24)
IV. E XPERIMENTAL S ETUP B. Notch Filtered Finite Time Control Consider the block diagram shown in Fig. 2, which corresponds to the finite-time controlled linearized system θ¨ = u, q¨ + 2ζωn q˙ + ωn2 q = −αu.
− s2+2ζωαns+ω2
q
n
θc +
| · |asign(·) −
v +
−
s2+2ζωns+ωn2 u s2+2ωns+ωn2
1 s
θ˙
1 s
We demonstrate the effectiveness of the controller using the flexible link setup shown in Fig. 3. This setup is comprised of a thin stainless steel link as the flexible appendage as and a servomotor. Tables I and II (Quanser.com) show the values of the flexible arm and servomotor parameters. To properly generate the torque profile developed by the controller, the input voltage to the motor must be derived to perform the slewing maneuver while suppressing the beam vibrations.
θ
| · |bsign(·)
Fig. 2.
Control system block diagram.
The output of the finite time controller is given by (with θc = 0 as the commanded attitude) ˙ b sign(θ), ˙ v = −|θ|a sign(θ) − |θ|
(17)
where b ∈ (0, 1), a > b/(2 − b) are controller parameters. ˙ = This feedback law controls the rigid body motion to (θ, θ) (0, 0) in finite time [12]. We use the finite-time control signal (17) as the input to the notch filter. The notch filter (16) can be expressed as u ¨ + 2ωn u˙ +
ωn2 u
= v¨ + 2ζωn v˙ +
ωn2 v.
To obtain a state-space realization of the filter define an auxiliary signal z satisfying z¨ + 2ωn z˙ + ωn2 z = v so that the transfer function from v to z is given by z 1 = 2 . v s + 2ωn s + ωn2 k,(((
Fig. 3.
The flexible link module setup (Quanser.com).
The torque developed by the motor is Tm = ηm ηg Kt Ia ,
(25)
where Kt is the motor torque constant and Ia is the armature current. The motor and gearbox efficiencies are represented by ηm and ηg , respectively. The differential equation for the armature circuit is then given by La I˙a + Ra Ia + Em = Vi ,
(26)
where Vi is the armature voltage (input to the servomotor), La is the armature inductance, Ra is the armature resistance,
and Em is the back electromotive force (EMF) voltage of the motor, which is defined as Em = Km θ˙m ,
(27)
where Km is the motor voltage constant and θ˙m is the angular velocity of the motor, which can be expressed in terms of the gear ratio Kg and the beam angular velocity as ˙ θ˙m = Kg θ.
(28)
The system torque is then given by (29)
τ = Kg Tm .
Using (25), (27)-(29), equation (26) can be re-written as La I˙a +
Ra ˙ τ = Vi − Km Kg θ. ηm ηg Kt Kg
Substituting these values into (12)-(14) we obtain ωn = k/mq = 22.39 rad/s, c = 2mq ζωn = 0.034 N · s/m, α = mθq /mq = 0.0802 m. Thus the nonlinear equations of motion given by (7)-(8) for the experimental setup can be written as 0.0069 + 0.758q 2 θ¨ + 0.0608¨ q + 1.516q q˙θ˙ = τ, q¨ + 0.045q˙ + 501.2q = −0.0802θ¨ + q θ˙2 . The torque and the filter equation can be expressed as
(30)
τ = (0.002 + 0.758q 2)(v − 44.74z) ˙ − 0.0027q˙ − 30.47q 2 ˙ ˙ + 1.516q q˙θ + 0.0608q θ ,
In general, the armature inductance La is very small and is therefore ignored in this paper. Consequently, equation (30) simplifies to τ=
ηm ηg Kt Kg ˙ (Vi − Km Kg θ). Ra
(31)
z¨ + 44.78z˙ + 501.2z = v, where, with a = b = 1/3, the finite time control law v is given by ˙ 1/3 sign(θ). ˙ v = −|θ|1/3 sign(θ) − |θ|
TABLE I F LEXIBLE ARM PARAMETERS Variable l0 l m ρl EI k c ζ
Name Attach pt. distance Arm length Beam mass Beam linear density Beam flexural rigidity Stiffness coefficient Dissipative constant Beam damping ratio
Value ≈0 0.483 0.065 0.1346 0.293 ≈0 0.0339 0.001
The voltage required by the motor then can be written in terms of the torque τ as
Unit m m kg kg/m N · m2 N · m/rad N · m/rad/s N/A
˙ Vi = 9.63τ + 0.46θ. A computer implementation of the filtered finite time control law was used to drive the system to the origin. The results of the computer simulation for a sample initial condition given by ˙ q, q) (θ, θ, ˙ = (π/4 rad, 0.1 rad/s, 0, 0)
TABLE II S ERVOMOTOR PARAMETERS Variable Ra La ηm ηg Kt Km Kg Iz Vmax
Name Armature resistance Armature inductance Motor efficiency Gearbox efficiency Motor torque constant Motor back EMF constant Gear ratio Equivalent moment of inertia Input voltage range
Value 2.60 2.18 0.69 0.85 0.00767 0.00767 60:1 1.8884 × 10−3 ±10
Unit Ω mH N/A N/A N·m V/rad/s N/A kg · m2 V
V. C OMPUTER S IMULATIONS This section illustrates the effectiveness of the filtered finite time control technique described in Section III through computer simulations. We use Tables I and II (Quanser.com) to compute the values of the constants used in the simulations. The values of mθq , mq , k, and It are computed using (2)-(5) as mθq = 0.0608 kg · m, mq = 0.758 kg, k = 379.94 N/m, It = 0.0069 kg · m2 . k,(((
and (z, z) ˙ = (0, 0) are shown in Figs. 4-6. It can be seen that the 45-degree slewing maneuver is achieved in about 5 seconds without any residual vibrations while the magnitude of the input voltage does not exceed 0.2 V. Clearly, the performance of the filtered finite time controller is analogous to that of “deadbeat”controllers for discrete time systems. VI. CONCLUSIONS We have revisited the slew maneuver control problem for flexible spacecraft. A filtered finite time control law has been proposed to achieve fast and precise slewing maneuvers without residual vibrations. The results have been applied to a benchmark flexible system and computer simulations have been included to illustrate the effectiveness of the control law. Future research includes experimental validation of the results of the paper. VII. ACKNOWLEDGMENTS The author wishes to acknowledge the support provided by Embry-Riddle Aeronautical University.
k,(((
0.8
θ (rad)
0.6 0.4 0.2 0 −0.2
0
5
10
15
20
15
20
t (s)
θ˙ (rad/s)
0.4 0.2 0 −0.2 −0.4
0
5
10
t (s)
Fig. 4.
Angular position and velocity.
−4
2
x 10
q (m)
1 0 −1 −2
0
5
10
q˙ (m/s)
2
15
20
15
20
t (s)
−3
x 10
1
0
−1
0
5
10
t (s)
Fig. 5.
Modal coordinate and velocity.
0.01
τ (N · m)
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0.005 0 −0.005 −0.01
0
5
10
15
20
15
20
t (s) 0.05 0
Vi (V)
R EFERENCES
−0.05 −0.1 −0.15 −0.2
0
5
10
t (s)
Fig. 6.
Input torque and voltage.
